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Phantasms of infinity

By

Julie J. Rehmeyer

Submitted by Marianne on July 2, 2012

This article first appeared on the FQXi community website, which does for physics and cosmology what Plus does for maths: provide the public with a deeper understanding of known and future discoveries in these areas, and their potential implications for our worldview. FQXi are our partners in our Science fiction, science fact project, which asked you to nominate questions from the frontiers of physics you'd like to have answered. This article addresses the question "Does infinity exist?". Click here to see other articles on the topic.

Infinity is a pain.
Its paradoxes easily ensnare the unsuspecting
reasoner. So over the centuries,
mathematicians have carefully constructed
bulwarks against its predations,
finally seeming to have it at bay.

But now cosmologists have developed
theories that put them squarely outside
the mathematicians' "green zone" of
safety. And as the mathematicians might
have predicted, infinity is having its way
with them. Spectacularly.

Raphael Bousso

For instance, imagine that, bang!, the
atoms in the room spontaneously coalesce
into a fully functioning brain next to
you. This odd scenario is really, really,
really unlikely — but not, quite, impossible,
as physicist Ludwig Boltzmann realised in
the nineteenth century. Boltzmann's farout
idea is now causing cosmologists fits,
because they have come to believe there
is an infinity of time and space. In that
case, "every stupid thing that can happen,
will happen, infinitely many times," says
Raphael Bousso, a cosmologist at the
University of California, Berkeley.

Cosmologists are not happy about this
state of affairs, but it gets even worse. A
real universe, as ours presumably is, may
be even less likely to develop than a
brain spontaneously blipping into existence.
Taken to an extreme, their calculations
suggest that we, and the entire
universe around us, might well just be a
dream in a Boltzmann brain.

But nobody seriously believes that.
Boltzmann brains are just phantasms,
cosmologists say, showing them that
they haven't gotten their mathematics
right. "Well, I hope we are not Boltzmann
brains," says Alexander Vilenkin, a
cosmologist at Tufts University. "But it's
hard to prove."

An explosion of universes

The problems began when many cosmologists came to believe that a fantastically fertile multiverse spews forth a mind-boggling profusion of infinitely many universes every second. If that isn't strange enough, each universe in the multiverse may have its own version of the laws of physics.

Alexander Vilenkin

Cosmologists have been pushed into
this incredible vision because it's the
only way they can make sense of the
mathematics that does such a good job of
explaining more mundane features of the
Universe, such as the Big Bang. But they
acknowledge that they don't have proof
yet. "This model hasn't been directly
tested the way people have tested electricity,
for example," says Sergei Winitzki of the Ludwig-Maximilians University
in Munich. "It's obviously not
easy to create a universe in the lab and
see how it expands."

Even without practical experiments, the
cosmologists have to find some way of
verifying their theory, or they're not doing
science. "The point of science is the more
we understand how things work, the better
we're able to predict," says Jaume Garriga
of the University of Barcelona.

One prize would be to predict the
value of the cosmological constant, a
number linked with the mysterious dark
energy that is pulling our Universe apart (see the Plus article Lambda marks the spot).
The theory, though, says that any value
is possible for the cosmological constant —
not a particularly helpful prediction.
And that's not all — the same thing
happens with most of the predictions
the cosmologists would like to make.

Sergei Winitzki

So cosmologists, trying another tack, reason this way. Suppose I want to predict what your height is. If I know that half the people in the world are between 5-foot-4- inches and 5-foot-8-inches, and less than 1 percent of people are taller than 6-foot-6 or shorter than 4-foot-6, I can guess that you are around 5-foot-6-inches tall and expect not be too far off. Further, if I find out that you're actually 12 feet tall, I might suspect that my initial information about probabilities was wrong.

Cosmologists would like to use similar
reasoning for physical properties like the
cosmological constant. If they calculate
the chance that the cosmological constant
will taken on any particular value in
a randomly chosen universe, then the
value of our cosmological constant will
probably be one of the popular choices.
"If the prediction doesn't match, then
we have to reevaluate our theories,"
says Winitzki.

Infinity's snares

So far, so good. But when cosmologists
try this strategy, they have to grapple
with infinitely many universes. And one
of the ways mathematicians keep themselves
out of trouble with infinity is that
they never, ever calculate probabilities
with infinitely many objects.

Jaume Garriga

To explain why, imagine that the multiverse
is like a weird bank with infinitely
many vaults, some containing bags of
money and some empty. To contain all
those vaults, not only does the bank
have infinitely long corridors, it also has
infinitely many floors. It's your lucky day:
you get the money in one vault, chosen
at random. So, what are the chances
you're going to strike it rich?

On the first floor, you notice that the
vaults with even numbers have money
and the odd ones don't — a 50-50 chance
of riches! But on the second floor, there's
money in only every tenth vault. On the
third floor, it's every fifth. You check the
first vault on each floor, but then you
discover that you hit it rich only every
hundredth room. Now you feel a bit faint
as your jackpot seems to fade.

Your confusion comes from a basic
mathematical conundrum. By choosing different ways to count, mathematicians
have shown, you can get any answer you
want. So really, you have no idea what
your chances of instant wealth are.

In the multiverse, each universe is like
a vault in the infinite bank, but instead of
bags of money, the universes have different
values of the cosmological constant.
And, just like your ill-fated venture in the
bank, cosmologists can get any answer
they want by choosing how they count.
"Without a way of calculating probabilities,
cosmology is a dead science, it doesn't
exist," says Vitaly Vanchurin of Tufts University.
"At first, I thought, this is crazy,
this is not science. But if we cannot answer
this question, we can't do cosmology."

Controlling the multiverse

Vitaly Vanchurin

One approach cosmologists are pursuing
to solve this problem is to figure out
which way of counting is the right one.
Then, they figure, they can count that
way only, and ignore the answers they
could get by counting some other way.
Another approach is to prune the universes
down to some finite size, thus
avoiding the infinity problem entirely.
Both approaches seems plausible, but
neither one has panned out yet.

Still, the cosmologists are optimistic
that they'll figure out a solution.
"There's a lot of evidence pointing at the
possibility that the universe really is
eternally inflating," Bousso says. "If that's
true, then surely there's a way of computing
probabilities. Don't ask me why,
but nature likes to be explainable."

Comments

A Boltzmann brain would seem to be rarer than a universe, to me anyway, a universe can be sort of a random blob with laws of physics conducive to evolving complex behavior, a Boltzmann brain has to go "SNAP I am a brain now!"
The average Universe probably never creates a Boltzmann Brain.
But there are still infinitely many, just like there are infinitely many integers.
For that matter, consider what happens if you notice that "I think therefore I am." applies to data about a Boltzmann brain. It doesn't have to be in any specific laws of physics other than numerical logic.
Then of course we point out that Not only does Quantum Mechanics demand a continuum number of universes based on our laws of physics, but Logic itself demands an infinite number of logically inaccessible universes with whatever self-consistent laws of physics they could possibly have, sans simulator.
So we can exist either as a simulated entity, or one that isn't simulated.
Which is more likely? I am gonna say the second due to the odds of random goop being googols of orders of magnitude more likely in any given rule-set than sentient beings randomly forming.
Think of it this way, Nobody ever proposed the theory of Random idiot design for why we exist.
On the other hand, while top-level Boltzmann brains would be exceedingly unusual and rare, top-level universes would produce brains that may act the same.

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